and can be interpreted in terms of a metric tensor \(g_{ij}\)  that provides lengths in the geometrical space of the \(\phi \)  fields, \(g_{ij}d\phi _id\phi _j\)  in any of the different coordinate choices.
This is the point of contact between physics experiments at colliders, i.e. production and scattering amplitudes derived from this Lagrangian, and the field geometry, where the former are represented in terms of curvature invariants. We now briefly recall the results presented in [1]}, [2]}.
